96

Irving Fisher—Mathematical investigations

just begin to buy, then his consumer’s rent or gain is (see Ch. I, § 8)

G = J' Vx yxdy — y l J' Vl x dy

yh Vk

or measuring this gain in the given commodity as valued at cents(say) per unit,

G

y.

-.n^J-xdy- f y

J Vk y > J V k

x dy.

This may be interpreted by a simple geometrical construction.In fig. 38 OA = y k and OR = y,.

Selecting the point 3 make the evident dotted construction deter-mining a point 5.

„ ., , 75 0 5 07 y „ [y \

Evidently: - = ^ ^ = - . •. 75 = (- . a: |

Let 3 assume all positions from M to 6. Then 5 will trace a curve26.

The area described by the moving line 73 is evidently J"' y ' xdy.The area described by the moving line 75 is evidentlyy/^ 1 ^— x 'jdy.

Hence area described by the moving line 35 is the difference ofthese expressions or G /y x .

That is the area M62 represents the gain measured in commodity.

Thus suppose a person buys corn measured by RAM6 and let cornat the valuation RO be the unit of utility. It is only the last layerR6 on which no gain is felt. For any preceding layer 75 the pricereally paid is OR while the price which it is worth to him is 07.The layer 73 may be considered as lengthened in his eyes by that